# How do you determine if rolles theorem can be applied to #f(x) = 2 − 20x + 2x^2# on the interval [4,6] and if so how do you find all the values of c in the interval for which f'(c)=0?

It is possible to apply and the answer is

The Rolles theorem says that if:

So:

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To determine if Rolle's Theorem can be applied to (f(x) = 2 - 20x + 2x^2) on the interval ([4, 6]), first check if the function satisfies the conditions of Rolle's Theorem:

- (f(x)) must be continuous on the closed interval ([4, 6]).
- (f(x)) must be differentiable on the open interval ((4, 6)).
- (f(4) = f(6)).

Now, calculate the derivative of (f(x)), denoted as (f'(x)). Then, find the critical points in the interval ((4, 6)) where (f'(c) = 0). These points represent the potential values of (c) where (f'(c) = 0).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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